UvA-DARE ( Digital Academic Repository ) Superconductivity on the Border of Weak Itinerant Ferromagnetism in UCoGe

Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: http://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible.

In the standard theory for superconductivity (SC) due to Bardeen, Schrieffer and Cooper ferromagnetic (FM) order impedes the pairing of electrons in singlet states [1]. It has been argued however that on the borderline of ferromagnetism, critical magnetic fluctuations could mediate SC by pairing the electrons in triplet states [2]. The discovery several years ago of SC in the metallic ferromagnets UGe 2 (at high pressure) [3], URhGe [4], and possibly UIr (at high pressure) [5], has put this idea on firm footing. However, later work provided evidence for a more intricate scenario in which SC in UGe 2 and URhGe is driven by a magnetic transition between two polarized phases [6][7][8], rather than by critical fluctuations associated with the zero temperature transition from a paramagnetic to a FM phase. Here we report a novel ambientpressure FM superconductor UCoGe. Since SC occurs right on the borderline of FM order, UCoGe may present the first example of SC stimulated by critical fluctuations associated with a FM quantum critical point (QCP).
UCoGe belongs to the family of intermetallic UTX compounds, with T a transition metal and X is Si or Ge, that was first manufactured by Troć and Tran [9]. UCoGe crystallizes in the orthorhombic TiNiSi structure (space group P nma ) [10,11], just like URhGe. From magnetization, resistivity (T ≥ 4.2 K) [9,10] and specific heat measurements (T ≥ 1.2 K) [12] it was concluded that UCoGe has a paramagnetic ground state. This provided the motivation to alloy URhGe (Curie temperature T C = 9.5 K) with Co in a search for a FM QCP in the series URh 1−x Co x Ge (x ≤ 0.9) [13]. Magnetization data showed that T C upon doping first increases, has a broad maximum near x = 0.6 (T max C = 20 K) and then rapidly drops to 8 K for x = 0.9 [13]. This hinted at a FM QCP for x 1.0. In this Letter we show that the end (x = 1.0) compound UCoGe is in fact a weak itinerant ferromagnet. Moreover, metallic ferromagnetism coexists with SC below 0.8 K at ambient pressure.
Polycrystalline UCoGe samples were prepared with nominal compositions U 1.02 CoGe (sample #2) and U 1.02 Co 1.02 Ge (sample #3) by arc melting the constituents (natural U 99.9%, Co 99.9% and Ge 99.999%), under a high-purity argon atmosphere in a water-cooled copper crucible. The as-cast samples were annealed for ten days at 850 • C. Samples for the different experiments were cut by spark erosion, after which the defected surface was removed by polishing. Powder X-ray diffraction patterns at T = 300 K confirmed the TiNiSi structure. The lattice constants extracted are a = 6.845Å, b = 4.206Å and c = 7.222Å, in agreement with literature [11]. The phase homogeneity of the annealed samples was investigated by electron micro-probe analysis. The matrix has the 1:1:1 composition and all samples contained a small amount (2%) of impurity phases.
The dc-magnetization was measured for temperatures T ≥ 2 K and magnetic fields B ≤ 5 T in a SQUID magnetometer. The demagnetizing factor of our samples is small (N ≈ 0.08) and corrections due to the demagnetizing field were neglected. Four-point low-frequency acresistivity and ac-susceptibility data were obtained using a phase-sensitive bridge in the range T = 0.02 − 8 K. The specific heat was measured using a semi-adiabatic method employing a mechanical heat switch on a sample with mass 3 g for T = 0.5 − 10 K and with a weak thermal link on a sample with mass 0.1 g for T = 0.1 − 1.0 K. Thermal expansion data were collected using a capacitance dilatometer for T = 0.23 − 8 K.
In Fig. 1a we show M as a function of T (obtained after field cooling). The inflection point in M (T ) at 3 K signals a FM transition with an unusually small ordered moment m 0 . At the lowest temperature (2 K) the transition is not complete yet, but from the curvature of in Fig. 1a). In the right-hand inset of Fig. 1 we show M measured at fixed T between 2 and 4.5 K in an Arrott plot (i.e. M 2 vs µ 0 H/M ). The isotherm that intersects the origin determines the Curie temperature T C . We extract T C = 3 K, in agreement with the M (T ) data. The FM transition at 3 K shows up as a broad peak in the ac-susceptibility χ ac (T ) (Fig. 1b) and a hump in the resistance R(T ) (Fig. 1b). The magnetic transition is a robust property, as M (T ), χ ac (T ) and R(T ) data taken on samples prepared from different batches almost coincide. The small ratio of m 0 to the effective moment (p ef f = 1.7 µ B [9]) shows UCoGe is a weak itinerant ferromagnet [14,15].
In Fig. 2 we show the specific heat c(T ) and the linear thermal expansion coefficient, α(T ) = L −1 dL/dT , around the magnetic transition. The transition width is large (∆T C ∼ 1 K). The relative change ∆(c/T C )/(c/T C ) assuming an ideal transition (see dashed line in Fig. 2a) is only 25 % and the magnetic entropy associated with the transition is small (0.3% of Rln2) as expected for a weak itinerant ferromagnet [14] with a small ordered moment. The linear term in the electronic specific heat γ amounts to 0.057 J/molK 2 , which indicates UCoGe is a correlated metal, but the electron interactions are relatively weak. In α(T ) the magnetic transition appears as a large negative contribution. The size of the ideal- (b) Thermal expansion coefficient α(T ) for sample #3. The large negative contribution below ∼ 5 K is due to FM order. The dashed line gives the idealized transition in α(T ) with ∆α = −1.1x10 −6 K −1 . The total relative length change ∆L/L = (L(0.23K) − L(T ))/L associated with magnetic order is obtained by integrating αmag(T ) (i.e. the difference between the experimental data and the linear term α = aT with a = 1.1x10 −7 K −2 expected in the absence of FM order) and amounts to +1.9x10 −6 . The peak below ∼ 0.6 K is the thermodynamic signature of the SC transition. In a field of 1 T, applied along the dilatation direction ∆L/L, the magnetic transition is smeared and SC is not resolved.
ized sharp step ∆α is −1.1x10 −6 K −1 at T C = 3 K (see dashed line in Fig. 2b) and presents a relative change ∆α/α of ≈ 3.3. This shows the magnetic transition is a bulk phenomenon.
Below 1 K UCoGe becomes superconducting as seen by a transition to zero in the resistance R(T ) and a large diamagnetic signal in χ ac (T ), see Fig. 1b and Fig. 3a. Unlike the magnetic properties, the SC properties depend sensitively on the quality of the samples as measured by the residual resistance ratio RRR = R(300K)/R(1K). For sample #2 (RRR = 10) and #3 (RRR = 25) SC is found with resistive onset temperatures of 0.61 K (Fig. 1b) and 0.82 K (Fig. 3a), respectively. In these polycrystalline samples the SC transition is relatively broad (∆T s ≈ 0.15 K). The in-phase component of the ac-susceptibility χ ac starts to drop when the resistive transition is complete. The drop is accompanied by a small dissipative peak in the out-of-phase signal χ ac (not plotted). At the lowest T the diamagnetic screening reaches a value of 60-70% of the ideal screening value χ M = -1/(1-N). This indicates UCoGe is a type II SC which is always in the mixed phase. A similar observation [4] with a compara- ble screening fraction was made for URhGe. Because of the intrinsic FM moments the local field is non-zero and the magnitude of χ ac is reduced.
Proof for bulk SC is obtained by specific heat (Fig. 3b) and thermal expansion measurements (Fig. 3c). The specific heat plotted as c/T versus T shows a broad transition with T onset s ≈ 0.66 K, which is almost equal to the temperature at which the resistance becomes zero. A rough estimate for the step-size of the idealized transition (dashed line in Fig. 3b) in the specific heat (at T s ≈ 0.45 K) is ∆(c/T s )/γ ≈ 1.0, which is smaller than for a conventional SC (the BCS value is 1.43), but comparable to the value [4] for URhGe. In the thermal expansion an equivalent broad SC transition is observed. Upon entering the SC state α(T ) shows a steady increase. We estimate the step-size ∆α ≈ 3.8x10 −7 K −1 , assuming an ideal sharp transition (see dashed line in Fig. 3c) at T s = 0.45 K. This step-size is comparable to the ones (with opposite sign) extracted from thermal expansion measurements on the heavy-fermion superconductors URu 2 Si 2 [16] and UPt 3 [17,18]. In a magnetic field of 1 T SC is suppressed and the thermodynamic signature of the transition is no longer resolved (see Fig. 3c). The α(T )-data also show that magnetism and SC coexist. The total relative length change associated with SC, obtained by integrating α sc (T ) after correcting for the normal-state linear contribution α = aT with a = −2.7x10 −7 K −2 (see dashed line for 0.45 K ≤ T ≤ 1 K in Fig. 3c) amounts to ∆L/L = −0.1x10 −6 and is small compared to the length change ∆L/L = +1.9x10 −6 due to magnetic ordering (see caption Fig. 3). Thus magnetism is not expelled below T s and coexists with SC.
In Fig. 4 we show the upper critical field B c2 (T ) for samples #2 and #3. The curvature (or tail) of B c2 is attributed to sample inhomogeneities. The quasi-linear behavior of B c2 (T ) at high fields extrapolates to SC transitions in zero field at 0.30 K and 0.60 K. These values are close to T onset s for bulk SC. From the slope dB c2 /dT and the values of γ and the residual resistivity ρ 0 , we can make a crude estimate [19] for the coherence length (ξ) and the mean free path ( ). For sample #3 dB c2 /dT = -5.2 T/K and ρ 0 = 12 µΩcm, and we calculate ξ ≈ 150Å and ≈ 500Å. This indicates sample #3 satisfies the clean-limit condition ( > ξ), a prerequisite for unconventional SC [20]. For the less pure sample #2 we find ξ ≈ 200Å and ≈ 300Å. The value of B c2 at the lowest T exceeds the BCS Pauli paramagnetic limit [19] (B P auli c2 = 1.8T s ≈ 1 T for sample #3), which for spin-singlet pairing is only possible in the case of strong spin-orbit scattering. On the other hand, the absence of Pauli limiting is expected for a triplet SC with equal-spin pairing state [21].
The small ordered moment of 0.03 µ B and low Curie temperature locate UCoGe close to the FM instability (i.e. the limit T C → 0). The proximity to the FM QCP can be further investigated using the Ehrenfest relation for second-order phase transitions dT C /dp = V m T C ∆α/∆c (with the molar volume V m = 3.13x10 −5 m 3 /mol). From the estimated step-sizes in α(T ) and c(T ) at T C we calculate dT C /dp = −0.25 K/kbar. This shows that the critical pressure p c at which magnetism vanishes is low (an upper-bound for p c assuming a linear suppression of T C is ∼ 12 kbar). In the same way we find that the SC transition temperature increases with pressure at a rate dT s /dp ≈ 0.048 K/kbar. In the scenario of the coexistence of p-state SC and FM [2], the increase of T s with pressure places UCoGe in the phase diagram on the far side of the SC lobe with respect to the critical point (compare UGe 2 at pressures of 10-12 kbar [3]). Accordingly, upon applying pressure, T s is predicted to pass through a maximum before vanishing at the magnetic critical point. The derived pressure dependencies of T C and T s for UCoGe have an opposite sign compared to those for URhGe. In URhGe T C shows a monotonic increase under pressures up to 120 kbar [22] and T s is suppressed with pressure. The positive pressure dependence of T s in UCoGe may explain the large difference in onset temperatures for superconductivity in the transport and bulk properties. Positive stress at the grain boundaries could cause a small volume fraction of the samples to have a larger T s .
The occurrence of SC in a FM material is naturally explained [2] by the formation of Cooper pairs with parallel spin. In UCoGe the proximity to the magnetic instability, the defect sensitivity of T s and the absence of Pauli limiting are all in agreement with such a scenario. Within the symmetry classification for orthorhombic itinerant FM spin-triplet superconductors [23] the SC gap is predicted to be anisotropic with point nodes along the magnetic moment direction or line nodes in the plane perpendicular to the moments. The determination of the gap function, however, requires experiments on single crystals. In the case of URhGe, which belongs to the same symmetry class as UCoGe, upper critical field measurements [24] on a single crystal indicate a p-wave polar order parameter with a maximum gap parallel to the a axis (the order moment points along the c axis [4]). The difference of a factor 7 in the size of the ordered moment m 0 (for URhGe the powder-averaged moment is m 0 ≈ 0.21µ B [4]) and the opposite pressure effects on T C and T s seem to indicate that UCoGe and URhGe represent two different cases of magnetically mediated SC. Indeed the recent observation of field-induced SC [8] in URhGe was taken as evidence for SC stimulated by a spin rotation in the neighborhood of a quantum phase transition under high magnetic field. In the case of UGe 2 the situation is again different as the FM to paramagnetic transition at the critical pressure becomes first order [3]. Moreover, evidence [6,7] is available that SC is driven by a changing Fermi surface topology associated with a metamagnetic jump in the magnetization. Consequently, unlike URhGe and UGe 2 , UCoGe may present a genuine case of SC at a FM quantum critical point.
In conclusion, we have demonstrated that UCoGe is a weak ferromagnet below T C = 3 K and becomes superconducting upon further cooling with T s = 0.8 K for the best sample. The sizeable discontinuities in the thermodynamic properties at both transition temperatures provide evidence for the bulk-like nature of both states. The coexistence of FM and SC is unusual and suggests SC mediated by magnetic interactions rather than by phonons. Since both SC and FM occur at ambient pressure, UCoGe offers a unique opportunity to elucidate the long-standing issue of SC stimulated by critical fluctuations associated with a magnetic quantum critical point.
This work was part of the research program of FOM (Dutch Foundation for Fundamental Research of Matter) and COST Action P16 ECOM. Funding by the Helmholtz Association under VH-VI 127 is gratefully acknowledged.